Question

find the perimeter of the polygon with the given vertices l(1, 4), n(2, 0), m(4, 0), p(-1, -2). the perimeter is about units.

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate distance $LP$

For points $L(1,4)$ and $P(-1,-2)$, $x_1 = 1,y_1 = 4,x_2=-1,y_2 = - 2$. Then $d_{LP}=\sqrt{(-1 - 1)^2+(-2 - 4)^2}=\sqrt{(-2)^2+(-6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10}\approx 6.32$.

Step3: Calculate distance $PM$

For points $P(-1,-2)$ and $M(4,0)$, $x_1=-1,y_1 = - 2,x_2 = 4,y_2 = 0$. Then $d_{PM}=\sqrt{(4+1)^2+(0 + 2)^2}=\sqrt{25 + 4}=\sqrt{29}\approx5.39$.

Step4: Calculate distance $MN$

For points $M(4,0)$ and $N(2,0)$, $x_1 = 4,y_1 = 0,x_2 = 2,y_2 = 0$. Then $d_{MN}=\sqrt{(2 - 4)^2+(0-0)^2}=\sqrt{(-2)^2}=2$.

Step5: Calculate distance $NL$

For points $N(2,0)$ and $L(1,4)$, $x_1 = 2,y_1 = 0,x_2 = 1,y_2 = 4$. Then $d_{NL}=\sqrt{(1 - 2)^2+(4 - 0)^2}=\sqrt{(-1)^2+16}=\sqrt{17}\approx4.12$.

Step6: Calculate perimeter

The perimeter $P=d_{LP}+d_{PM}+d_{MN}+d_{NL}\approx6.32+5.39+2+4.12 = 17.83$.

Answer:

$17.83$